Firstly, if you haven’t completed the survey, you can still do it here. All it does is show you some pictures of sandwiches, and asks you to count them. Just because we have 1000 responses doesn’t mean we have enough data, and we would still love to hear your response!
If you still haven’t filled out the form, you should note that many of the respondents testify to the deep satisfaction of contributing to the important research in this form, as evidenced by the 10.7% of the written comments which feature the f-word.
Well, without further ado, let’s dive into the data.
Here are the responses to the central question: ‘does cutting a sandwich in half yield one sandwich or two?’
We have a clear winner, calling it one sandwich is more popular, and it is statistically significantly more popular. There were however still plenty of people who called it two sandwiches.
We’re going to call the respondents that said 1 the first camp, and the respondents that said 2 the second camp.
The people in the second camp mostly had the same reasons: they were looking at the shape of the individual components and deciding that because it had filling between two bits of bread, each component is a sandwich. (One respondent was working with shape but said ‘Any isolated piece of bread is a sandwich’)
In the first camp, however, there were a couple of different ways people reached their conclusion. Some were looking at the shape of the sandwich at time of construction, saying that cuts were irrelevant.
Others in the first camp were going off amount of substance. A sandwich needs two full slices of bread. A small (but very vocal in the comments) minority gave weights to each component, so bread = 0.25, filling = 0.5, and they provided fractional answers to a lot of the later questions. Most people, however, were either looking at just the amount of bread, or just the amount of filling.
One interesting observation is how similar all these justifications looked to each other, here are some of the comments people left, each with a different view:
- ‘A sandwich is filling sandwiched between two full slices of bread.’
- ‘A sandwich is two pieces of bread separating the chosen additive, any cuts thereafter are irrelevant — it is *a* sandwich.’
- ‘My answers were based on the definition of “two slices of bread with stuff between them.”’
They’re all the same! Everyone is agreed that a sandwich is ‘two slices of bread with stuff between them’, and yet this definition led the first two to say “one sandwich” where the third says “two sandwiches”.
Moving on to the other questions, there was pretty much no difference in counting between the horizontal and diagonal cut, as expected.
A surprising number changed their mind when the other half was taken away, however. Presentation matters to a lot of people. Here is a quote from someone in the second camp who said half:
‘If I have a sandwich which was cut in half so that I have 2 now, and if I keep both halves, I consider this 2 sandwiches. If I have one half of a sandwich I refer to it as half a sandwich, to better describe it.’
Similarly a good number of people in the first camp decided that in the absence of the other half of the sandwich, there was nothing it would be half of, and so it must be a whole sandwich.
There were few surprises when cutting it into four. I can’t, however, work out what was going on in the minds of the 61 defectors. There was a lot of what appeared to be random noise in the data from people just putting anything in the box, but 61 feels like a lot of them.
There were similarly few surprises when it came to deciding whether soldiers were one or four. It is interesting though, that more people in the second camp didn’t want to call soldiers sandwiches.
Most people called the double-decker one sandwich. You can see in the first camp that 36 people belonged to the group that weighted bread and filling. (e.g. if bread = 0.25, and filling = 0.5, then the double-decker is 1.75)
People had different ways of justifying that this was one sandwich, one said ‘Inner layers of bread are still fillings.’ Another used ‘Any isolated piece of bread is a sandwich.’
We come to the open-faced sandwich, which was a controversy in itself, and one independent of camp. Roughly the same proportions of the first and second camps decided that the open-faced sandwich was not in fact a sandwich.
People are significantly happier to call a folded sandwich a sandwich than they are to call the open-faced sandwich a sandwich. There are still some, however, who insist that ‘A sandwich is two *separate* pieces of bread, (or whatever) With substance in the middle, no matter how many times you cut it’
One thing I find really interesting: look at the proportion of people who gave fractional answers for the unfolded and folded versions of this sandwich.
In the first camp, folding increases the number of people who gave it a fraction, because they were saying that the unfolded one was not a sandwich, but now that it is a sandwich it must be a fraction. (Some people pointed out that this sandwich has twice as much filling as the first half-sandwich)
In the second camp, however, folding decreases the number of people who gave it a fraction, because now that it’s folded, it’s a full sandwich.
Now we come to the double-fold. Note that the end result of this one looks almost identical to the original question. Yet, here, calling it two sandwiches is a clear winner.
The second camp had no problems here, but in the first camp, a very small number picked up on the fact that it looks like the first question and said one. A very large proportion of them, however, decided that this was in fact two sandwiches. This adds a lot of weight to the time-of-construction theory.
This is the half-eaten sandwich. Note again, this looks very similar to an early question, except it is a little rougher around the edges.
The first camp answered consistently with the other time we had half a full sandwich. The second camp however have been divided. Some of them stick to their definition of any meat separated by bread. Others have been influenced by the fact we call it a half-eaten sandwich and have switched to calling it a half.
I did not expect Tubby Toast to give us anything interesting, it was included purely as a childhood TV reference. The first camp fell in line, but a good 82 of the second camp defected.
One person in the second camp told me that they switched because the smiley face made it seem like the sandwich was complete.
The mobius sandwich was actually not my idea, it was suggested by the second comment in the survey, and it was a genius idea. It drives a wedge between those that say ‘in the mobius strip sandwich there’s only one slice of bread … therefore the answer is 0’ and those that say ‘Sandwiches should contain the rest of the meal and be held between a grasped hand’
Some people thought that this was a diagram of a four-dimensional hypersandwich, because of its resemblance to the common projection of a hypercube. This was not my intention, this is a three-dimensional cube of filling surrounded on all faces with bread. If it were a hypersandwich projection, one could rotate the meat from the inside to the outside, making it an open sandwich.
Some people didn’t want to call the cube a sandwich, and others did, but perhaps the most interesting answer was three. These people decided that the sandwich cube was three orthogonal sandwiches at once.
When I began debating with a friend over whether we were looking at one or two sandwiches, I had no idea that we not only had different answers to the simple question, but that we had different worldviews that cover whole field of counting sandwiches that play out differently in many different sandwich circumstances.
In conclusion, as one commenter said: ‘Remember that all of these are facts about language and cognition, not about sandwiches. Sandwiches don’t care what numbers are assigned to them.’
I’d like to thank everyone who helped share the survey, especially Artifexian who featured it on a podcast episode.
If you would like to contribute further to the endeavour to count sandwiches, there are more questions here.
